For the data in the table, does y vary directly with x? If it does, write an equation for the direct variation.
x y
4 6
8 12
10 15
slope between 1st and 2nd = (12-6)/(8-4) = 6/4 = 3/2
slope between 1st and 3rd = (15-6)/(10-4) = 9/6 = 3/2
slope between 2nd and 3rd = (15-12)/(10-8) = 3/2
mmhhh?
To determine if y varies directly with x, we need to check if the ratio between y and x remains constant for all the given data.
In this case, the ratio y/x is 6/4 = 1.5 for the first data pair, 12/8 = 1.5 for the second data pair, and 15/10 = 1.5 for the third data pair.
Since the ratio remains constant at 1.5 for all data points, we can conclude that y varies directly with x.
To find the equation for the direct variation, we need to express it in the form y = kx, where k is the constant of variation.
To determine the value of k, we can pick any data point and substitute the values of x and y into the equation. Let's use the first data point (x = 4, y = 6):
6 = 4k
Solving for k, we divide both sides of the equation by 4:
k = 6/4
k = 1.5
Therefore, the equation for the direct variation is y = 1.5x.
To check if y varies directly with x, we need to see if there is a constant ratio between y and x when comparing the values in the table.
Let's calculate the ratios of y to x for each pair of values:
- For the first pair (4, 6), y/x = 6/4 = 1.5
- For the second pair (8, 12), y/x = 12/8 = 1.5
- For the third pair (10, 15), y/x = 15/10 = 1.5
Since the ratios are the same for all pairs, we can conclude that y varies directly with x.
To find the equation for direct variation, we can write it in the form y = kx, where k is the constant of variation.
From the table, we can see that the ratio of y to x is always equal to 1.5. Therefore, the equation for direct variation is:
y = 1.5x